But if we look at how the share of the vote between the three big parties has
changed in each constituency from 2001 to 2005, we see a different picture.

Each cross represents a constituency in England, Scotland or
Wales.

A cross's position shows how the share of the vote won by the
three big parties was split between them in that constituency.

A cross in the red zone usually means a Labour win, but in a
few cases the seat was won by a fourth party (e.g. the SNP) while
Labour got more votes than Conservative and Liberal Democrat.
Similarly for crosses in the orange and blue zones.

You can interpret the movement of seats as two-way swing by
referring to this diagram:

Outside the triangle of black lines there is no argument about
which of the three big parties voters preferred.
Inside are seats where the winner got
less than 50 per cent of the 3-way vote. With a different voting
system like approval or alternative voting, these seats could have
been won by a different party.

Are you seeing what I'm seeing? Upward not sideways?

The dominant swing is from Labour to Lib Dem, even in Tory seats.

But there are many exceptions, with vote shares moving in all
directions in the Conservative and Lib Dem zones.

On the boundaries (marginals)...

Some Conservative wins from Labour are because of
Lab to Con swing but more rely on Lab to LD swing.

There is little movement in the Labour vote near the
Con / Lib Dem boundary.

So anti-Tory tactical
voting may have helped in some Labour seats but not
others, and seems to have done nothing for the Lib Dems.
Anti-war tactical
voting may have contributed to the outcome, but its
effects are buried in the overall Lab to LD movement.

Most of the handful of seats moving right to left are Scottish
constituencies where the comparison between 2001 and 2005 is
imprecise because of boundary changes. There is very little
swing towards Labour anywhere.

The major shift is towards the centre of a three-party map.
If you believe in voting reform, the votes inside the black
triangle are those you don't trust to represent people's real
preferences. Their number is increasing.

The clustering of votes in the centre is even clearer if we compare
Labour, Conservative and Other (which includes the Lib Dems, Nationalists
and Independents). Here is the current picture, with a significant majority
of seats inside the black triangle.

Most ways of summing up elections are two-way: "Swing", "X majority over
Y". But most elections are multi-way and the overall picture is more
complicated than just the average of lots of little pictures. A
multi-way visualization is valuable, not only because many people support third,
fourth and fifth parties, but also because it shows the electorate is not
a homogeneous lump who "wanted a Labour government with a reduced majority,
and that's what the election gave them", which seems to be the pundits'
favourite analysis.

A three-way visualization doesn't tell the full multi-party story, but it's a
step in the right direction. The pictures above (for 2001) were the first I'd
seen. After a Google search turned up only the 3D swingometer on Wikipedia,
I tried to do better and came up with the triangular map. But someone at
the BBC had beaten me to it: Peter Snow had one on Election 2005 -- and
a much more beautiful one than mine. Peter's didn't really show the reality
of a 3-party contest though -- he kept calling it a battleground and fixated on
the fronts, where the battles were all two-way, as usual. People turned
a different colour when they "fell in battle" instead of moving.
It's no surprise that the BBC also didn't have the black
inner triangle. Pointing out how many contests have questionable outcomes
doesn't have the drama of winner-takes-all conflicts. But it would allow
Peter to do some interesting what-ifs if we had an alternative vote system:

Peter: What if all those non-Labour voters in Labour
seats were prepared to use their second preference
votes to get Labour out at any cost?

Peter: But what if they were saying, let's give Labour
a kicking but not let the Tories in?

The first example assumes that Labour voters don't want either Con
or LD in non-Labour seats (reflecting the Labour/non-Labour antagonism of
Peter's hypothetical question), whereas the second example assumes Labour
voters share the "don't let the Tories in" philosophy so prefer LD in
non-Labour seats. Those arguments may be a little too
nuanced for Election night TV, but they're plausible, and they show how
people's preferences really can affect outcomes under the alternative vote.

Some more reasons to adopt the alternative vote rather than
first-past-the-post are given
here.